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This question is inspired by reading Arnold Neumaier's article on Everett's measurement theory:
http://arnold-neumaier.at/physfaq/topics/everett
Suppose I have a system whose state lies in a two-dimensional Hilbert space, and I want to model a measurement of its state as a unitary evolution. As I understand it, the standard way to do this is to first find a basis for the system's Hilbert space; let's call the two basis states ##\vert \psi \rangle## and ##\vert \chi \rangle##. Then we have to model the system doing the measurement; this system will have to have a three-dimensional Hilbert space with basis states ##\vert R \rangle##, ##\vert \Phi \rangle##, and ##\vert X \rangle##, where ##\vert R \rangle## is the "ready" state and ##\vert \Phi \rangle## and ##\vert X \rangle## are the states that indicate that ##\vert \phi \rangle## and ##\vert \chi \rangle##, respectively, have been measured.
Then the measurement process is modeled by assuming that the Hamiltonian of the total system, which I won't write down explicitly, induces the following:
First, for ##\vert \Phi \rangle## and ##\vert X \rangle## to be proper "indicator" states, we must have
$$
\vert \phi \rangle \vert \Phi \rangle \rightarrow \vert \phi \rangle \vert \Phi \rangle
$$
$$
\vert \chi \rangle \vert X \rangle \rightarrow \vert \chi \rangle \vert X \rangle
$$
i.e., both of these states are left invariant by the Hamiltonian (because the measuring system already indicates the "correct" state for the measured system). Then, for the measuring system to actually make a measurement, we must have
$$
\vert \phi \rangle \vert R \rangle \rightarrow \vert \phi \rangle \vert \Phi \rangle
$$
$$
\vert \chi \rangle \vert R \rangle \rightarrow \vert \chi \rangle \vert X \rangle
$$
i.e., the "ready" state gets changed to the appropriate "indicator" state by the measurement.
However, it seems like there is a problem with the above evolution: it can't be unitary, because it's not invertible! Note that the states ##\vert \phi \rangle \vert \Phi \rangle## and ##\vert \chi \rangle \vert X \rangle## appear on the RHS of *two* separate evolution rules, with *different* states on the LHS. That means that, for example, if we find the system in state ##\vert \phi \rangle \vert \Phi \rangle##, if the above rules are correct, we can't tell whether that state arose from ##\vert \phi \rangle \vert \Phi \rangle## in the past, i.e., because the system was just sitting there unchanged, or whether it arose from ##\vert \phi \rangle \vert R \rangle## in the past, i.e., because a measurement took place.
So it seems like there's no way to model what we usually think of as a "measurement" by a unitary operator. Am I missing something?
http://arnold-neumaier.at/physfaq/topics/everett
Suppose I have a system whose state lies in a two-dimensional Hilbert space, and I want to model a measurement of its state as a unitary evolution. As I understand it, the standard way to do this is to first find a basis for the system's Hilbert space; let's call the two basis states ##\vert \psi \rangle## and ##\vert \chi \rangle##. Then we have to model the system doing the measurement; this system will have to have a three-dimensional Hilbert space with basis states ##\vert R \rangle##, ##\vert \Phi \rangle##, and ##\vert X \rangle##, where ##\vert R \rangle## is the "ready" state and ##\vert \Phi \rangle## and ##\vert X \rangle## are the states that indicate that ##\vert \phi \rangle## and ##\vert \chi \rangle##, respectively, have been measured.
Then the measurement process is modeled by assuming that the Hamiltonian of the total system, which I won't write down explicitly, induces the following:
First, for ##\vert \Phi \rangle## and ##\vert X \rangle## to be proper "indicator" states, we must have
$$
\vert \phi \rangle \vert \Phi \rangle \rightarrow \vert \phi \rangle \vert \Phi \rangle
$$
$$
\vert \chi \rangle \vert X \rangle \rightarrow \vert \chi \rangle \vert X \rangle
$$
i.e., both of these states are left invariant by the Hamiltonian (because the measuring system already indicates the "correct" state for the measured system). Then, for the measuring system to actually make a measurement, we must have
$$
\vert \phi \rangle \vert R \rangle \rightarrow \vert \phi \rangle \vert \Phi \rangle
$$
$$
\vert \chi \rangle \vert R \rangle \rightarrow \vert \chi \rangle \vert X \rangle
$$
i.e., the "ready" state gets changed to the appropriate "indicator" state by the measurement.
However, it seems like there is a problem with the above evolution: it can't be unitary, because it's not invertible! Note that the states ##\vert \phi \rangle \vert \Phi \rangle## and ##\vert \chi \rangle \vert X \rangle## appear on the RHS of *two* separate evolution rules, with *different* states on the LHS. That means that, for example, if we find the system in state ##\vert \phi \rangle \vert \Phi \rangle##, if the above rules are correct, we can't tell whether that state arose from ##\vert \phi \rangle \vert \Phi \rangle## in the past, i.e., because the system was just sitting there unchanged, or whether it arose from ##\vert \phi \rangle \vert R \rangle## in the past, i.e., because a measurement took place.
So it seems like there's no way to model what we usually think of as a "measurement" by a unitary operator. Am I missing something?
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